Statistics, including averages, for lengths of metal–ligand bonds are reported, together with some intraligand distances, for complexes of the d- and f-block metals. Mean values are presented for 325 different bond types involving metal atoms bonded to H, B, C, N, O, F, Si, P, S, Cl, As, Se, Br, Te or I atoms of the ligands.

9.6.1. Introduction

The determination of molecular geometry is of vital importance to our understanding of chemical structure and bonding. The majority of experimental data have come from X-ray and neutron diffraction, microwave spectroscopy, and electron diffraction. Over the years, compilations of results from these techniques have appeared sporadically. The first major compilation was Chemical Society Special Publication No. 11: Tables of Interatomic Distances and Configuration in Molecules and Ions (Sutton, 1958). This volume summarized results obtained by diffraction and spectroscopic methods prior to 1956; a supplementary volume (Sutton, 1965) extended this coverage to 1959. Summary tables of bond lengths between carbon and other elements were also published in Volume III of International Tables for X-ray Crystallography (Kennard, 1962). Some years later, the Cambridge Crystallographic Data Centre (Allen, Bellard, Brice, Cartwright, Doubleday, Higgs, Hummelink, Hummelink-Peters, Kennard, Motherwell, Rodgers & Watson, 1979) produced an atlas-style compendium of all organic, organometallic and metal-complex crystal structures published in the period 1960–1965 (Kennard, Watson, Allen, Isaacs, Motherwell, Pettersen & Town, 1972). More recently, a survey of geometries determined by spectroscopic methods (Harmony, Laurie, Kuczkowski, Schwendemann, Ramsay, Lovas, Lafferty & Maki, 1979) has extended coverage in this area to mid-1977. A notable compendium of structural data, without geometric information, was given in Comprehensive Organometallic Chemistry (Bruce, 1981), covering all complexes with metal–carbon bonds. The BIDICS (Brown, Brown & Hawthorne, 1982) series, which finished in 1981, provided for some years a full coverage of metal complexes giving both bibliographic and geometric information. There have also been valuable annual summaries, without geometric information, on the structures of organometallic compounds determined by diffraction methods (Russell, 1988).

The production of further comprehensive compendia of X-ray and neutron diffraction results has been precluded by the steep rise in the number of published crystal structures, as illustrated by Fig. 9.6.1.1.
Print compilations have been effectively superseded by computerized databases. In particular, the Cambridge Structural Database now contains bibliographic, chemical, and numerical results for some 86 000 organo-carbon crystal structures. This machine-readable file fulfils the function of a comprehensive structure-by-structure compendium of molecular geometries. However, the amount of data now held in the CSD is so large that there is also a need for concise, printed tabulations of average molecular dimensions.

Growth of the Cambridge Structural Database as number of entries (Nent) added annually. The structures containing d- or f-block metals are indicated by shading.

The only tables of average geometry in general use are those contained in the Chemical Society Special Publications of 1958 and 1965 (Sutton, 1958, 1965), which list mean bond lengths for a variety of atom pairs and functional groups. Since these early tables were based on data obtained before 1960, we have used the CSD to prepare a new table of average bond lengths in organic compounds (see Chapter 9.5
) and in metal complexes. The table given here (Table 9.6.3.3) specifically lists average lengths for metal–ligand distances, together with intra-ligand distances, involving bonds between the d- and f-block metals (Sc–Zn, Y–Cd, La–Hg, Ce–Lu, Th–U) and atoms H, B, C, N, O, F, Si, P, S, Cl, As, Se, Br, Te, and I of ligands. Mean values are presented for 324 different bond types involving such metal–ligand bonds.

9.6.2. Methodology

9.6.2.1. Selection of crystallographic data

All results given in Table 9.6.3.3 are based on X-ray and neutron diffraction results retrieved from the September 1985 version of the CSD. Neutron diffraction data only were used to derive mean bond lengths involving hydrogen atoms. This version of the CSD contained results for 49 854 single-crystal diffraction studies of organo-carbon compounds; 9802 of these satisfied the acceptance criteria listed below and were used in the averaging procedures:

(i) Structure contains a d- or f-block metal.

(ii) Atomic coordinates for the structure have been published and are available in the CSD.

(iii) Structure was determined from diffractometer data.

(iv) Structure does not contain unresolved numeric data errors from the original publication (such errors are usually typographical and are normally resolved by consultation with the authors).

(v) Only structures of higher precision were included on the basis that either (a) the crystallographic R factor was ≤ 0.07 and the reported mean estimated standard deviation (e.s.d.) of the C—C bond lengths was ≤ 0.030 Å (corresponds to AS flag = 1, 2 or 3 in the CSD), or (b) the crystallographic R factor ≤ 0.05 and the mean e.s.d. for C—C bonds was not available in the database (AS = 0 in the CSD).

(vi) Where the structure of a given compound had been determined more than once within the limits of (i)–(v), then only the most precise determination was used.

The structures used in Table 9.6.3.3 do not include compounds whose structure precludes them from the CSD (i.e. not containing `organic' carbon). In practice, structures including at least one C—H bond are taken to contain `organic' carbon. Thus, the entry for Cr—CO distances has a contribution from [NEt4][Cr(μ-H)(CO)10] but not from K[Cr(μ-H)(CO)10] or [Cr(CO)6].

9.6.2.2. Program system

All calculations were performed on a University of Bristol VAX 11/750 computer. Programs BIBSER, CONNSER, RETRIEVE (Allen et al., 1979) and GEOSTAT (Murray-Rust & Raftery, 1985a,b), as locally modified, were used. A stand-alone program was written to implement the selection criteria, whilst a new program (STATS) was used for statistical calculations described below. It was also necessary to modify CONNSER to improve the precision with which it locates chemical substructures. In particular, the program was altered to permit the location of atoms with specified coordination numbers. This was essential in the case of carbon so that atoms with coordination numbers 2, 3, and 4 (equivalent to formal hybridization state sp1, sp2, sp3) could be distinguished easily and reliably. Considerable care was taken to ensure that the correct molecular fragment was located by GEOSTAT in the generation of geometrical tabulations. Searches were conducted for all metals together and statistics for individual metal elements and subdivision of the entry for a given metal carried out subsequently. An important modification to GEOSTAT allowed for calculation of metal-atom coordination number with due allowance for multihapto ligands and μ2 ligands. Thus, η5-C5H5, η6-C6H6, and other η5 and η6 ligands were assigned to occupy 3 coordination sites, η3 and η4 ligands such as allyls and dienes to occupy 2 coordination sites, and η2 ligands such as alkenes 1 site, and so on. The approach taken in dealing with (μ2) bridging ligands was that when a metal–metal bond is bridged by one atom of a ligand [e.g. as in Cl, CO, OMe etc. as in (a), (b) below] then only the non-metal atom is counted as occupying a coordination site. For the relatively rare case of bridging polyhapto ligands (in which the bridging atoms are linked by direct bonds), the assignment follows logically, thus, μ2-η2,η2-alkyne, see (c) below, occupies one site on each metal. Bridging ligands that do not have one atom bonded to both metals [e.g. acetate in (d) below] contribute to metal coordination numbers as do terminal ligands. In examples (a)–(d) below, the metal atoms therefore have coordination numbers as follows: (a), Rh 4; (b), Fe 6; (c), Co 4; (d), Rh 6. For cases where coordination number is very difficult to assign, notably where a metal atom is bonded to more than one other metal atom as in metal cluster complexes, no assignment was attempted.

The non-location of hydrogen atoms presents major difficulties, both in the determination of coordination numbers for metal atoms, and for correct identification of ligands (e.g. to distinguish methoxide from methanol). Care was therefore taken to exclude cases where any ambiguity existed [e.g. no data taken for M—(OCH3) and M—O(H)CH3 distances when both are present in a structure in which hydrogen-atom positions were not reported].

9.6.2.3. Classification of bonds

The classification of metal–ligand bonds in Table 9.6.3.3 is based on the ligating contacting atom. Thus, all metal–boron distances appear in sections 2.1–2.3 of Table 9.6.3.3, all metal–carbon distances in sections 3.1–3.22, and so on. Where intra-ligand interatomic distances (e.g. P—C distances in tertiary phosphines) are given in Table 9.6.3.3, they are averaged over all metals and precede the individual metal–ligand interatomic distances for that ligand.

Table 9.6.3.3 is designated: (i) to appear logical, useful, and reasonably self-explanatory to chemists, crystallographers, and others who may use it; (ii) to permit a meaningful average value to be cited for each bond length. With reference to (ii), it was considered that a sample of bond lengths could be averaged meaningfully if: (a) the sample was unimodally distributed; (b) the sample standard deviation (σ) was reasonably small, ideally less than ca 0.04 Å; (c) there were no conspicuous outlying observations – those that occurred at > 4σ from the mean were automatically eliminated from the sample by STATS, other outliers were inspected carefully; (d) there were no compelling chemical reasons for further subdivision of the sample. It should be noted that Table 9.6.3.3 is not intended to be complete in covering all possible ligands. The purpose of the table is to provide information on the interatomic distances for ligands of the greatest chemical importance, notably for those that are simple and/or common.

9.6.2.4. Statistics

Where there are less than four independent observations of a given bond length, then each individual observation is given explicitly in Table 9.6.3.3. In all other cases, the following statistics were generated by the program STATS.

(i) The unweighted sample mean, d, where and is the ith observation of the bond length in a total sample of n observations. Recent work (Taylor & Kennard, 1983, 1985, 1986) has shown that the unweighted mean is an acceptable (even preferable) alternative to the weighted mean, where the ith observation is assigned a weight equal to 1/var(di). This is especially true where structures have been pre-screened on the basis of precision.

(ii) The sample median, m. This has the property that half of the observations in the sample exceed m, and half fall short of it.

(iii) The sample standard deviation, σ, where

(iv) The lower quartile for the sample, . This has the property that 25% of the observations are less than and 75% exceed it.

(v) The upper quartile for the sample, . This has the property that 25% of the observations exceed and 75% fall short of it.

(vi) The number (n) of observations in the sample.

The statistics given in Table 9.6.3.3 correspond to distributions for which the automatic 4σ cut-off (see above) had been applied, and any manual removal of additional outliers (an infrequent operation) had been performed. In practice, a very small percentage of observations were excluded by these methods. The major effect of removing outliers is to improve the sample standard deviation, as shown in Fig. 9.6.2.1(b)
in which four (out of 366) observations are deleted.

Effects of outlier removal and subdivision based on coordination number and oxidation state. Cu—Cl: (a) all data; (b) all data without outliers [> 4σ (sample) from mean]; (c) all data for which Cu is 4-coordinate, CuII.

The statistics chosen for tabulation effectively describe the distribution of bond lengths in each case. For a symmetrical, normal distribution, the mean (d) will be approximately equal to the median (m), the lower and upper quartiles () will be approximately symmetric about the median , and 95% of the observations may be expected to lie within ±2σ of the mean value. For a skewed distribution, d and m may differ appreciably and and will be asymmetric with respect to m. When a bond-length distribution is negatively skewed, i.e. very short values are more common than very long values, then it may be due to thermal-motion effects; the distances used to prepare the table were not corrected for thermal libration.

In a number of cases, the initial bond-length distribution was clearly not unimodal as in Fig. 9.6.2.1(a). Where possible, such distributions were resolved into their unimodal components (as in Fig. 9.6.2.1c) on chemical or structural criteria. The case illustrated in Fig. 9.6.2.1, for Cu—Cl bonds, is one of the most spectacular examples, owing to the dramatic consequences of oxidation state and coordination number (and Jahn–Teller effects) on the structures of copper complexes.

9.6.3. Content and arrangement of table of interatomic distances

Table 9.6.3.1 indicates how the interatomic distances covered in Table 9.6.3.3 are subdivided. Metal–ligand distances are grouped according to the ligand contact atom, which leads to ordering by atomic number of that contact atom. For a given contact atom (H, B, C, etc.), the ligands are grouped by type as listed in Table 9.6.3.1. The class of ligand is identified numerically (e.g. alkoxides are class 5.3, alcohols class 5.23, ethers 5.24, etc.). Particular ligands are identified by a third number (e.g. methoxide is ligand 5.3.1). Finally, alternative bonding modes for a particular ligand are denoted by a fourth number [e.g. terminal alkoxides 5.3.1.1, bridging (μ2) alkoxides 5.3.1.2]. In general, the bonding modes are arranged in the sequence , , etc., where ηn implies n atoms of the ligand are bonded to metal atoms, and μm that m metal atoms are bonded to the ligand. Thus, acetates are represented by entries headed 5.5.2.1 (η1), 5.5.2.2 (chelating, η2) and 5.5.2.3 (bridging, μ2). For each ligand, the metal–ligand bonds then follow a sequence of ascending atomic number of the metal. For a given metal, the first line of an entry in Table 9.6.3.3 gives statistics covering all appropriate occurrences of metal–ligand distances. Further lines give statistics for metal–ligand distances for subdivision based largely on chemical criteria (e.g. metal oxidation state or coordination number). Cases where one atom of a ligand bridges two or more metal atoms were included only when the metal atoms were all of the same type and, unless specified, only when the metal–ligand distances were symmetrical (range for distances ≤ 0.1 Å).

In many instances, the number of structures having inter-atomic distances involving a given metal for a particular ligand is too small (< 4) for statistics to be quoted. In these cases, individual structures, and the distances in them, are given. These structures are identified by their CSD reference code (e.g. BOZMIN); short-form literature references, ordered alphabetically by reference code, are in Appendix 9.6.2.

Each line of Table 9.6.3.3 contains nine columns of which six record the statistics of the bond-length distribution described above. The content of the remaining three columns: Bond, Substructure, and Note, are described below.

9.6.3.1. The `Bond' column

This specifies the atom pair to which the line refers. Therefore, in the case of triethylphosphine complexes (section 8.5.2), there are 18 lines, in which the bond column contains P—C, followed by 17 entries for Ti—P through to Au—P, indicating statistics for both intraligand and metal–ligand atom pairs.

9.6.3.2. Definition of `Substructure'

This column provides details of any subdivision of particular metal–ligand bonds that has been applied. Thus, for terminal iron–chlorine bonds (in section 10.1.1.1), the second and third lines of the Fe—Cl entry refer to complexes in which the iron atom is four-coordinate and in oxidation state II and III, respectively. In some cases, subdivision has been carried out on the basis of ligand substituents in those cases where a well defined subdistribution was observed. For clarity, in a number of cases the ligand structure and numbering scheme are illustrated in Fig. 9.6.3.1.
The reader will be aware that formal oxidation state is not always well defined, where no assignment was possible then this is indicated by (−) rather than the roman numeral used elsewhere. Finally, cases where the ligand oxidation state is variable are identified (e.g. for O2, o-quinones etc.) by references to the footnotes at the end of Table 9.6.3.3.

9.6.3.3. Use of the `Note' column

The `Note' column refers to the footnotes collected in Appendix 9.6.1. These record additional information as follows: (a) notable features of the distribution of distances, e.g. likely bias due to dominance by one structure of substructure, skewness, bimodality (subdivisions of the entry usually follow, which remove these features whenever possible); (b) further details of the chemical substructure, such as the exclusion of structures with particular trans ligands; (c) details of exclusion criteria used for a given entry or group of entries, such as the constraint that the two M—Cl distances, in bridging (μ2) chloride complexes, differ by < 0.1 Å (section 10.1.1.2); (d) references to previously published surveys of crystallographic results relevant to the entry in question. We do not claim that these entries are in any way comprehensive and we would be grateful to authors for notification (to AGO) of any omissions. This will serve to improve the content of any future version of Table 9.6.3.3.

9.6.3.4. Locating an entry in Table 9.6.3.3

Table 9.6.3.2 provides a `guide' to the contents of Table 9.6.3.3. The number of entries for which individual examples of M—ligand distances are quoted, and the number of entries for which statistics are given in Table 9.6.3.3 are listed for each metal. Inspection of Table 9.6.3.2 shows which element pairs have no bond lengths recorded in Table 9.6.3.3. Thus, while there are no cobalt–fluorine distances in Table 9.6.3.3, there are 6 classes of cobalt–phosphorus distances for which there are examples quoted and 10 for which statistics are given.